3,156 research outputs found
Geometric structure in smooth dual and local Langlands conjecture
No full textThis expository paper first reviews some basic facts about p-adic fields, reductive p-adic groups, and the local Langlands conjecture. If G is a reductive p-adic group, then the smooth dual of G is the set of equivalence classes of smooth irreducible representations of G. The representations are on vector spaces over the complex numbers. In a canonical way, the smooth dual is the disjoint union of subsets known as the Bernstein components. According to a conjecture due to ABPS (Aubert–Baum–Plymen–Solleveld), each Bernstein component has a geometric structure given by an appropriate extended quotient. The paper states this ABPS conjecture and then indicates evidence for the conjecture, and its connection to the local Langlands conjecture
Reduced C*-Algebra of the p-Adic Group GL(n) II
No full textThe reduced C*-algebra of the p-adic group GL(n) admits a Bernstein decomposition. We give a minimal refinement of this decomposition, and provide structure theorems for the reduced Iwahori–Hecke C*-algebra and the reduced spherical C*-algebra. This leads to a very explicit description of the tempered dual of GL(n) in terms of Bernstein parameters and extended quotients. We also prove that Plancherel measure (on the tempered dual of a reductive p-adic group) is rotation-invarian
On the Weyl character formula for SU(n)
No full textIn this Note, we draw a straight line between the representation theory of SU(3) and the SU(3)-classification schemes in particle physics. Our approach is based on that of Weyl, but we have in mind the versions which appear, “in modern dress,” in Adams and Bott. Our formulation brings an important part of particle physics into line with two contemporary accounts of compact Lie groups
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